I'm trying to create a simulation of a three pendulum rotary harmonograph, the one you can see in action in this video or in these instructions.

As you can see in the video, there are 2 pendulum with 1 degree of freedom each (one axis of movement) plus 1 pendulum with 2 degrees of freedom (the one in the middle, holding the sheet of paper).

I'm currently able to simulate the movement of a pen attached to just 2 pendulum (with 1 degree of freedom each) using this formula

(t = time, f = frequency, p = phase, d = damping, a = amplitude)

The result of this formula will give me the position of the pen on the x axis. I can then use it a second time with different values to obtain the position of the pen on the y axis. Very straightforward: result is here.

So, how can I add the third pendulum (the one with 2 degrees of freedom)? This formula can compute the position of a rod connected to a pendulum with two degrees of freedom:

However, I don't think this is what I need. The formula only gives me as a result one number: the movement of the pendulum makes the sheet of paper move on two axis (and I would even say 3, judging by the fact that the sheet of paper doesn't stay parallel to the plane).

So how in the world do I project the movement of a pendulum moving in 3d space over a 2d plane? Is this what I really need to do? If yes how?

Finally, even assumed that I'm able to do so, how should I insert these results into the pen x and y position? I presume is a matter of just making an addition: am I wrong?

(p.s. please forgive any possible triviality in my question. As you probably have guessed, I've never been taught physics nor computer science)

EDIT:

here's my pain. I can only simulate figures on the left. The one on the right requires a third pendulum, and this third pendulum is not oscillating in just one direction but two (or three?!).. (Image from "Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology").

+1 for introducing me to an amazingly awesome physical system that I'd never heard of before and what is now one of my favorite YouTube videos!
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joshphysicsFeb 12 '13 at 0:21

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Yep, I'm enjoying the "artistic" part of physics. Glad you like it... Try to play around with the simulator I've made. Just change the frequency of one of the pendulum to a multiple of the other and look at it go :) Hope someone may help to put in a third rotatory pendulum!
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SaturnixFeb 12 '13 at 0:23

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The 3-D to 2-D mapping isn't really a problem. While your system will move in three dimensions, you can reduce the problem to one of 2 coordinates, which greatly improves the solvability of the system. An equation of constraint gives you the mapping between your 2 dynamical coordinates and your 3 spatial dimensions. (For example, imagine a marble racing around on the surface of a paraboloid; the marble is moving in 3 dimensions, but it obeys an equation of constraint: $z = x^2 + y^2$).
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KDNFeb 18 '13 at 12:22